Suppose that the functions s and t are defined for all real numbers x as follows s(x) x-4 and t(x)=4x+1 write the expressions for (s•t) and (x) and (s+t) and evaluate (s-t)(-2)
(s•t)(x) = s(t) = t-4 = (4x+1)-4 = 4x-3
so, (s•t)(-2) = 4(-2) - 3 = -11
Or, s(t(-2)) = s(-7) = -7-4 = -11
(s+t)(x) = s(x)+t(x) = x-4 + 4x+1 = 5x-3
do s-t similarly, and plug in -2 for x
To find the expressions for (s • t), (x), and (s + t), we need to perform the specified operations on the given functions.
1. (s • t):
This represents the composition of functions s and t. We can find it by substituting the function t(x) into function s(x):
s • t = s(t(x)) = s(4x + 1)
2. (x):
The expression (x) signifies that we need to evaluate the function s(x) at a specific value of x. In this case, we don't have any constraint, so we can use any valid real number.
3. (s + t):
To add the two given functions, s(x) and t(x), we simply combine the corresponding terms:
s + t = (x - 4) + (4x + 1)
Now we can proceed to evaluating (s - t)(-2).
To do that, we substitute -2 into the expression (s - t):
(s - t)(-2) = s(-2) - t(-2) = (-2 - 4) - (4 * (-2) + 1)
Simplifying the above expression, we get:
(s - t)(-2) = -6 - (-7) = -6 + 7 = 1
Therefore, (s - t)(-2) equals 1.
To express the composite function (s∘t)(x), we substitute t(x) into s.
(s∘t)(x) = s(t(x))
= s(4x+1)
= (4x+1) - 4
= 4x - 3
The expression for (s+t)(x) can be obtained by adding s(x) and t(x):
(s+t)(x) = s(x) + t(x)
= (x-4) + (4x+1)
= 5x - 3
To evaluate (s-t)(-2), we substitute -2 into both s(x) and t(x):
(s-t)(-2) = s(-2) - t(-2)
= (-2 - 4) - ((4 * -2) + 1)
= -6 - (-7)
= 1